Bernstein set

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In mathematics, a Bernstein set is a subset of the real line that meets every uncountable closed subset of the real line but that contains none of them.{{Cite book

|last=Oxtoby

|first=John C.

|title=Measure and Category

|edition=2nd

|year=1980

|page=24}}

A Bernstein set partitions the real line into two pieces in a peculiar way: every measurable set of positive measure meets both the Bernstein set and its complement, as does every set with the property of Baire that is not a meagre set.{{citation|title=Point Set Theory|volume=131|series=Chapman & Hall/CRC Pure and Applied Mathematics|first=John C. II|last=Morgan|publisher=CRC Press|year=1989|isbn=9780824781781|page=163|url=https://books.google.com/books?id=WwmvxtDlz9UC&pg=PA163}}.